I started my save using GoG's browser demo of the game and then imported the save game to the full GoG Windows desktop version after finishing the demo.

Now I am in Game Mode+ and I tried upgrading the Charlatan's Staff in my inventory but it wouldn't.
I checked in my inventory and found I already have an upgraded "Charlatan's Staff+" but the original version is still there too.

I tried to fix it by opening the save game with Notepad++ and removing the entire line

<item _class="weapon" type="staff" level="3" owner="" alt="0"/>

but once I load the save game the staff gets auto-added by the game!

How can I fix this?

Phew.

Does it prevent getting the all NewGame+ weapons achievement thing?

Nope, because all the bug does is give you an extra, un-upgraded Charlatan's Staff. The only reason you can't upgrade it is because you already have an upgraded one. You should still be able to grab the achievement.